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Triangle of generalized Eulerian numbers T(n,k) = <n,k>_2 read by rows, n >= 1, 0 <= k < 2*n.
7

%I #22 Feb 22 2024 20:18:31

%S 1,1,1,1,1,4,11,4,1,1,11,72,114,72,11,1,1,26,367,1492,2438,1492,367,

%T 26,1,1,57,1630,13992,48965,73120,48965,13992,1630,57,1,1,120,6680,

%U 109538,727982,2169674,3107640,2169674,727982,109538,6680,120,1

%N Triangle of generalized Eulerian numbers T(n,k) = <n,k>_2 read by rows, n >= 1, 0 <= k < 2*n.

%C T(n,k) is the number of nonnegative integer n X n matrices with every row and column sum 2 and sum of entries below the main diagonal k. The case when every row and column sum is 1 is given by the Eulerian numbers (A008292). - _Andrew Howroyd_, Feb 22 2020

%H Andrew Howroyd, <a href="/A269742/b269742.txt">Table of n, a(n) for n = 1..1600</a> (first 40 rows)

%H Esther M. Banaian, <a href="http://digitalcommons.csbsju.edu/honors_thesis/24">Generalized Eulerian Numbers and Multiplex Juggling Sequences</a>, (2016). All College Thesis Program. Paper 24.

%H E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, <a href="http://arxiv.org/abs/1508.03673">A generalization of Eulerian numbers via rook placements</a>, arXiv:1508.03673 [math.CO], 2015.

%H Andrew Howroyd, <a href="/A269742/a269742.txt">PARI Program</a>

%e Triangle begins:

%e 1;

%e 1, 1, 1;

%e 1, 4, 11, 4, 1;

%e 1, 11, 72, 114, 72, 11, 1;

%e 1, 26, 367, 1492, 2438, 1492, 367, 26, 1;

%e 1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1;

%e ...

%e The matrices for row n=3, k=0..2 are:

%e [2 0] [1 1] [0 2]

%e [0 2] [1 1] [2 0]

%o (PARI) \\ See link. - _Andrew Howroyd_, Feb 22 2020

%Y Row sums are A000681.

%Y Columns k=0..4 are A000012, A000295, A260585, A260575, A260582.

%Y Central coefficients are A332729.

%Y Cf. A008292, A269743, A269744.

%K nonn,tabf

%O 1,6

%A _N. J. A. Sloane_, Mar 16 2016

%E Terms a(26) and beyond from _Andrew Howroyd_, Feb 22 2020